Optimal. Leaf size=284 \[ -\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {15 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {b c d^2 \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {9 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.68, antiderivative size = 315, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5798, 5740, 5685, 5683, 5676, 30, 14, 266, 43} \[ -\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5}{4} c^2 d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {15 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {b c^5 d^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {9 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d^2 \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 43
Rule 266
Rule 5676
Rule 5683
Rule 5685
Rule 5740
Rule 5798
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+c^2 x\right )^2}{x} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (-2 c^2+\frac {1}{x}+c^4 x\right ) \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {9 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {15 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.83, size = 305, normalized size = 1.07 \[ \frac {1}{128} d^2 \left (240 a c \sqrt {d} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+\frac {16 a \left (2 c^4 x^4-9 c^2 x^2-8\right ) \sqrt {d-c^2 d x^2}}{x}+64 b c \sqrt {d-c^2 d x^2} \left (\frac {2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}-\frac {2 \cosh ^{-1}(c x)}{c x}\right )+\frac {32 b c \sqrt {d-c^2 d x^2} \left (2 \cosh ^{-1}(c x)^2+\cosh \left (2 \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}-\frac {b c \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.57, size = 550, normalized size = 1.94 \[ -\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a \,c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a \,c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a \,c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a \,c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {15 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d^{2} c}{16 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) d^{2}}{\left (c x +1\right ) \left (c x -1\right ) x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{5} x^{4}}{16 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {9 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{3} x^{2}}{16 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c \,\mathrm {arccosh}\left (c x \right )}{\sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d^{2} c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{6} \mathrm {arccosh}\left (c x \right ) x^{5}}{4 \left (c x +1\right ) \left (c x -1\right )}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{4} \mathrm {arccosh}\left (c x \right ) x^{3}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{2} \mathrm {arccosh}\left (c x \right ) x}{8 \left (c x +1\right ) \left (c x -1\right )}-\frac {33 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c}{128 \sqrt {c x +1}\, \sqrt {c x -1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, {\left (10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d x + 15 \, \sqrt {-c^{2} d x^{2} + d} c^{2} d^{2} x + 15 \, c d^{\frac {5}{2}} \arcsin \left (c x\right ) + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{x}\right )} a + b \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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